# Recent questions in Linear algebra

Vectors and spaces

### An observational study is retrospective if it considers only existing data. It is prospective if the study design calls for data to be collected as time goes on. Tell which of the following observational studies are retrospective and which are prospective. In a study of post-traumatic stress disorder, soldiers who have been in combat are given biannual physical and psychological tests for five years after they return from active duty.

Forms of linear equations

### Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named $$x_1, x_2,...x,...$$ ,… from left to right.

Forms of linear equations

### Let $$AX = B$$ be a system of linear equations, where A is an $$m\times nm\times n$$ matrix, X is an n-vector, and $$BB$$ is an m-vector. Assume that there is one solution $$X=X0$$. Show that every solution is of the form $$X0+Y$$, where Y is a solution of the homogeneous system $$AY = 0$$, and conversely any vector of the form $$X0+Y$$ is a solution.

Vectors and spaces

### Is there a relationship between Facebook use and age among college students? The following two-way table displays data for the 219 students who responded to the survey. \begin{array}{ccc} & \text{Age} \ \text{Facebook user?} & {\begin{array}{l|c|c|c} & \begin{array}{c} \text { Younger } \ (18-22) \end{array} & \begin{array}{c} \text { Middle } \ (23-27) \end{array} & \begin{array}{c} \text { Older } \ (28 \text { and up)} \end{array} \ \hline \text { Yes } & 78 & 49 & 21 \ \hline \text { No } & 4 & 21 & 46 \end{array}} \ \end{array} What percent of the students who responded were older Facebook users?

Forms of linear equations

Linear algebra

### Determine whether the statement is true or false. Justify your answer. When using Gaussian elimination to solve a system of linear equations, you may conclude that the system is inconsistent before you complete the process of rewriting the augmented matrix in row-echelon form.

Forms of linear equations

### The reduced row echelon form of a system of linear equations is given.Write the system of equations corresponding to the given matrix. Use x, y; or x, y, z; or $$x_1,x_2,x_3,x_4$$ as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. [ 1 0 4 4 0 1 3 2 0 0 0 0 ]

Forms of linear equations

### Determine if the statement is true or false, and justify your answer. (a) Different sequence s of row operations can lead to different reduced echelon forms for the same matrix. (b) If a linear system has four equations and seven variables, then it must have infinitely many solutions.

Forms of linear equations

### The row echelon form of a system of linear equations is given. (a)Writethesystemofequationscorrespondingtothegivenmatrix.Usex,y;orx,y,z;or x_1,x_2,x_3,x_4 asvariables. (b)Determinewhetherthesystemisconsistentorinconsistent.Ifitisconsistent,givethesolution. \left[ 10215−1 \right]

Forms of linear equations

### Write the vector form of the general solution of the given system of linear equations. x1+4x4=0 x2−2x4 ​

Forms of linear equations

### Write the vector form of the general solution of the given system of linear equations. $$x_1+2x_2−x_3+x_4=0$$

Forms of linear equations

### In the matrix of a system of linear equations, suppose that two of the rows are equal. What can you say about the rowreduced form of the matrix?

Forms of linear equations

### Determine whether the statement is true or false. If the last row of the reduced row echelon form of the augmented matrix of a system of linear equations has only one nonzero entry, then the system is inconsistent.

Vectors and spaces

### The reduced row echelon form of the augmented matrix of a system of linear equations is given. Tell whether the system has one solution, no solution, or infinitely many solutions. Write the solutions or, if there is no solution, say the system is inconsistent.

Forms of linear equations

### $$x+y+2z=3$$ $$3x+2y+2z=4$$ $$x+y+3z=5$$

Forms of linear equations

### Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named $$x_1,x_2,...$$ form left to right. $$\begin{bmatrix}1&0&5&3&2\\0&1&-2&4&-7\\0&0&1&1&3\end{bmatrix}$$

Forms of linear equations

### When solving a system of linear equations using matrices, do you prefer to place the augmented matrix in row echelon form or in reduced row echelon form? Give reasons for your choice.

Forms of linear equations

### The reduced row echelon form of the augmented matrix of a system of three linear equations in three variables must be of the form $$\begin{bmatrix}1 & 0 & 0 & a \\ 0 & 1 & 0 & b \\ 0 & 0 & 1 & c \end{bmatrix}$$ where a, b, c, are real numbers. Justify your answer.

Forms of linear equations